\section{Controlling negative diffusion in the presence risk behavior changes}
\label{sec:intro.risk}
The study in Section~\ref{sec:intro.game} assumes that the behavior of
each individual remains the same before and after taking
interventions. However, this is not an accurate assumption in some
real world scenarios. For instance, people expose themselves more to
the public after taking vaccinations. This behavior change is often
referred as risk behavior change. And it is very common, specially in
epidemiology. Since vaccination is not 100\% reliable, this kind of
behavior change has the potential to increase the likelihood of
disease transmission. In our study, it is important to consider the
impact of risk behavior to good intervention strategies. In our
discussion below, we use disease transmission as example, but risk
behavior is not limited to epidemiology.

For many diseases, such as influenza and HIV, prophylactic
interventions using anti-virals and vaccinations are commonly used to
control the spread of the diseases, and are usually universally
recommended, barring individual constraints. Recent studies have shown
significant benefits of anti-retrovirals for reducing the spread of
HIV \cite{karim:science10}.  Such treatments have varying levels of
efficacy (25-75\% in the case of HIV \cite{karim:science10,gray:hiv}
and between 10-80\% in the case of influenza \cite{cdc-ve}, depending
on the demographics and the specifics of the flu strain).  However,
people are not very well aware of this limitation, and studies often
over-estimate the efficacy of vaccines~\cite{petrie:fluVaccine}.
Indeed, the perceived protection from infection might cause behavioral
changes, leading to an increase in contact by a treated individual;
such a behavioral change following vaccination could also be a natural
evolutionary response~\cite{klein:behaviorChange,moore:behavior}, and
has also been documented recently in the context of flu
vaccines~\cite{reiber:behaviorChange}.  Regardless of the underlying
reasons, failure of prophylactic interventions in conjunction with
increased social behavior can have significant unexpected effects on
the disease dynamics.  In a series of important
papers \cite{blower+risk94,blower+imperfect93} Blower and her
collaborators demonstrated that risk behavior change, in the context
of HIV vaccinations, could lead to perverse outcomes.  Subsequently,
several independent studies have confirmed this phenomenon of
perversity in the use of HIV vaccines and anti-virals, and vaccines
for the human papillomavirus
(HPV)~\cite{smith+b:hiv04,bezemer08,wilson+cwb,velasco,gray:hiv,vanderStraten:hiv,andersson:hiv,crosby:aids,halloran:hiv,meil:hiv,brewer:hpv,forster:hpv}.\footnote{
The phenomenon of an increase in risky behavior following protection
is also referred to as ``moral hazard'' and has been studied
extensively in a number of areas, such as insurance
(e.g., \cite{mirrlees99}); in the epidemiology literature, this is
referred to more commonly as ``risk behavior''
(e.g., \cite{blower+risk94}), and we will fix on this terminology for
most of the thesis.}

A fundamental question in mathematical epidemiology is to determine
the fraction of the population that needs to be vaccinated or treated
with anti-virals in order to minimize the impact of the disease,
especially when the supply is limited.  Modern epidemiological
analysis is largely based on an elegant class of models, called SIR
(susceptible-infected-recovered), which was first formulated by Reed
and Frost in the 1920s, and developed over the subsequent decades.
The SIR model and its variants have been highly influential in the
study of epidemics
\cite{yang+h1n109,manski+partial10,medlock+optvacc09,kaplan+smallpox02,grassly+model08,blower+imperfect93}. These
models, however, do not attempt to capture the rich structure of the
contact network over which interactions occur. Network structure has a
direct effect on both the spread of diseases as well as the nature of
interactions, which has been observed by a number of researchers,
e.g. \cite{newman:netstructure03, jackson10}. In the emerging area of
contact network epidemiology, an underlying contact graph captures the
patterns of interactions which lead to the transmission of a disease
\cite{pastor+episcalefree,dezso+virus02,lloyd+m:epidemiology,meyers06,meyers+sars05,newman:spread02,wang+xhcw:scalefree09,ganesh05}.
Many studies have predicted the spread of diseases through networks
using mathematical analysis or simulations.  As we have argued above,
moral-hazarding/risky behavior clearly plays an important role in the
effectiveness of such interventions.  While the impact of risky
behavior on prophylactic treatments has been studied in previous work,
the extent of the perversity and its dependence on network structure
as well as the precise nature of the behavior change has remained
largely unknown. \footnote{ Similar issues arise in the context of the
  spread of malware through infected computers.  Several studies,
  e.g., \cite{malware-link}, have found that computer and smart-phone
  users do not relate bot infections to risky behavior, such as
  downloading spam mails, though a large fraction of users have
  updated anti-virus software. It is plausible that such phenomena can
  also be associated with risky behavior in many cases.}

In this thesis, we study the impact of risk behavior change on the
spread of diseases in networks and observe a rich and complex
structure dependent both on the underlying network characteristics as
well as the nature of the change in behavior.  We use a discrete-time
SIR model of disease transmission on a contact network.  The contact
network is an undirected graph with each edge having a certain
probability of disease transmission. An infected node is assumed to
recover in one time step.  We consider both uniform random vaccination
(where each node is vaccinated independently with the same
probability) as well as targeted vaccinations (where nodes are
vaccinated based on their degree of connectedness). Vaccines are
assumed to fail uniformly and randomly.\footnote{Though we focus on
vaccinations and disease transmission, the basic results apply to
other prophylactic treatments such as anti-virals, and other phenomena
such as malware spread.}  We model risk behavior change by an increase
in the disease transmission probability. A significant aspect of our
work is the consideration of the ``sidedness'' of risk behavior
change. We classify risk behavior as one-sided or two-sided based on
whether the increase in disease transmission probability requires an
increase in risk behavior of both the infector and the infectee or
just the infector. As examples: influenza (H1N1) may be modeled as a
one-sided disease since a vaccinated individual may be motivated to
behave more riskily (going to crowded places, traveling on planes
etc.,) thus increasing the chance of infecting all the individual
comes in contact with; whereas AIDS (HIV) may be modeled as a
two-sided disease since the increase in disease transmission requires
both the individuals participating in the interaction to engage in
risky behavior. Of course, these examples are simplistic and most
diseases have elements of both one-sided as well as two-sided risk
behavior.  

Our main findings are threefold.  
\begin{itemize}
\item First, we find that the {\em severity of the epidemic varies
  non-monotonically as a function of the vaccinated fraction}.  The
specific dynamics depend on the nature of risky behavior, as well as
the efficacy of the vaccine (the less reliable the vaccine, the
greater the non-monotonicity) and the contagiousness of the disease,
but in general, we observe that increased vaccination does not
immediately imply less severity; in some cases, the severity could
increase by as much as a factor of two.
\item Second, we find that {\em one-sided risk behavior change leads
  to perverse outcomes at low levels of vaccination, while two-sided
  risk behavior change leads to perverse outcomes at high levels of
  vaccination}.  Our analysis indicates that effective prophylactic
  interventions against diseases with one-sided risk behavior change
  need to have sufficiently high coverage; on the other hand, for
  diseases with two-sided risk behavior change, it is essential to
  combine prophylactic treatments with education programs aimed at
  reducing risky behavior.
\item Our third and, perhaps, most surprising finding is that {\em
  interventions that target highly connected individuals can be
  strictly worse than random interventions} for the same level of
  coverage and that this phenomenon occurs both for one-sided as well
  as two-sided risk behavior change. Given prior work on targeting
  vaccine distributions, this finding flies in the face of intuition
  that expects that targeted vaccination would confer greater
  benefits.
\end{itemize}

Our results have direct implications for public policy on containing
epidemic spread through prohylactic interventions.  Implications of
risk behavior in public health have been examined earlier,
e.g. \cite{blower+risk94,blower+imperfect93}.  These prior studies are
based on differential equation models, which divide the population
into a fixed set of groups and model the interaction between different
groups in a uniform way. The epidemic spread is then characterized by
the ``reproductive number,'' denoted by $R_0$, with the expected
epidemic size exhibiting a threshold behavior in terms of $R_0$.  In
contrast, we use a network model that captures the fine structure of
interactions between (an arbitrary number of) individuals rather than
(a fixed set of) groups, and find that the network structure has a
significant impact on the resulting dynamics.  The heterogenous
network model extends to a larger range of real-life situations but
the increased fidelity comes at a price.  The outcomes are more
complicated and varied and the general approach of lowering $R_0$ does
not appear to be directly applicable.  Another new contribution of our
study is the focus on the sidedness inherent in risk behavior change,
which has not been considered before.  Prior research has implicitly
assumed one-sided risk behavior change where vaccinated individuals
engage in risky behavior increasing the chances of infection of those
they come in contact with. This work explicitly treats both one-sided
and two-sided risk behavior changes and shows that their differing
impact needs to be considered in public intervention policies.

\iffalse
\item One-sided vs.\ two-sided. 

We find that both forms of risk behavior change can lead to perverse
  outcomes across a wide range of contact networks. However, 
Our findings have implications for public policy and the distribution
of vaccines.  At
  a minimum it is essential to consider both the ``sidedness'' of risk
  behavior change as well as the achievable levels of vaccine coverage
  before taking action.

\item Random vs.\ targeted. We find that the previous findings
  regarding one-sided and two-sided risk behavior change apply
  irrespective of whether the immunization is uniformly random or
  targeted based on the node's degree. 

\end{itemize}
\fi



